Changing object location in local coordinates don't work
From http://en.wikibooks.org/wiki/Blender_3D:_Noob_to_Pro/Parenting:


  The “Clear Parent Inverse” function sets this inverse transformation to the identity transformation, so the child picks up the full parent object transformation.


I guess instead of adding the object to the parent with CTRLP, I have to do it so with CTRLSHIFTP, so that the child+parent relation ship is set and the Clear Parent Inverse is performed. 

NOTE: I don't understand the details so if anyone adds a proper answer I'll mark it as answer. For now though this will be the answer for this question.